Geometry & Topology

Hyperbolic jigsaws and families of pseudomodular groups, I

Beicheng Lou, Ser Peow Tan, and Anh Duc Vo

Full-text: Access by subscription


We show that there are infinitely many commensurability classes of pseudomodular groups, thus answering a question raised by Long and Reid. These are Fuchsian groups whose cusp set is all of the rationals but which are not commensurable to the modular group. We do this by introducing a general construction for the fundamental domains of Fuchsian groups obtained by gluing together marked ideal triangular tiles, which we call hyperbolic jigsaw groups.

Article information

Geom. Topol., Volume 22, Number 4 (2018), 2339-2366.

Received: 15 November 2016
Revised: 23 June 2017
Accepted: 9 October 2017
First available in Project Euclid: 13 April 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F06: Structure of modular groups and generalizations; arithmetic groups [See also 20H05, 20H10, 22E40] 20H05: Unimodular groups, congruence subgroups [See also 11F06, 19B37, 22E40, 51F20] 20H15: Other geometric groups, including crystallographic groups [See also 51-XX, especially 51F15, and 82D25] 30F35: Fuchsian groups and automorphic functions [See also 11Fxx, 20H10, 22E40, 32Gxx, 32Nxx] 30F60: Teichmüller theory [See also 32G15]
Secondary: 57M05: Fundamental group, presentations, free differential calculus 57M50: Geometric structures on low-dimensional manifolds

pseudomodular killer intervals hyperbolic jigsaw marked ideal triangle


Lou, Beicheng; Tan, Ser Peow; Vo, Anh Duc. Hyperbolic jigsaws and families of pseudomodular groups, I. Geom. Topol. 22 (2018), no. 4, 2339--2366. doi:10.2140/gt.2018.22.2339.

Export citation


  • O Ayaka, A computer experiment on pseudomodular groups, Master's thesis, Nara Women's University (2015)
  • D Fithian, Congruence obstructions to pseudomodularity of Fricke groups, C. R. Math. Acad. Sci. Paris 346 (2008) 603–606
  • H M Hilden, M T Lozano, J M Montesinos-Amilibia, A characterization of arithmetic subgroups of ${\rm SL}(2,{\mathbb R})$ and ${\rm SL}(2,{\mathbb C})$, Math. Nachr. 159 (1992) 245–270
  • D D Long, A W Reid, Pseudomodular surfaces, J. Reine Angew. Math. 552 (2002) 77–100
  • B Lou, S P Tan, A D Vo, Hyperbolic jigsaws and families of pseudomodular groups, II: Integral jigsaws, generalized continued fractions and arithmeticity, in preparation
  • G A Margulis, Discrete subgroups of semisimple Lie groups, Ergeb. Math. Grenzgeb. 17, Springer (1991)
  • H Proskin, Flat faces in punctured torus groups, Rocky Mountain J. Math. 37 (2007) 2025–2051
  • K Takeuchi, A characterization of arithmetic Fuchsian groups, J. Math. Soc. Japan 27 (1975) 600–612
  • J W Tan, The study of pseudomodular groups and the computations, Undergraduate honours thesis, National University of Singapore (2016)